A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation

Monthly Weather Review - Tập 148 Số 10 - Trang 3973-3994 - 2020
Pierre Tandeo1,2, Pierre Ailliot1, Marc Bocquet3, Alberto Carrassi4,5,6, Takemasa Miyoshi2, Manuel Pulido4,7,8, Yicun Zhen1
1Lab-STICC_IMTA_CID_TOMS (Technopole Brest Iroise CS 83818 29232 BREST cedex 3 - France)
2RIKEN CCS - RIKEN Center for Computational Science [Kobe] (7 Chome-1-26 Minatojima Minamimachi, Chuo Ward, Kobe, Hyogo 650-0047, Japon - Japan)
3d CEREA Joint Laboratory École des Ponts ParisTech and EDF R&D, Université Paris-Est, Champs-sur-Marne, France
4Department of Meteorology, University of Reading, Reading, United Kingdom
5Utrecht Mathematical Institute (PO Box 80125, 3508 TC Utrecht - Netherlands)
6f National Centre for Earth Observation, University of Reading, Reading, United Kingdom
7h Universidad Nacional del Nordeste, Corrientes, Argentina
8i CONICET, Corrientes, Argentina

Tóm tắt

AbstractData assimilation combines forecasts from a numerical model with observations. Most of the current data assimilation algorithms consider the model and observation error terms as additive Gaussian noise, specified by their covariance matrices and , respectively. These error covariances, and specifically their respective amplitudes, determine the weights given to the background (i.e., the model forecasts) and to the observations in the solution of data assimilation algorithms (i.e., the analysis). Consequently, and matrices significantly impact the accuracy of the analysis. This review aims to present and to discuss, with a unified framework, different methods to jointly estimate the and matrices using ensemble-based data assimilation techniques. Most of the methods developed to date use the innovations, defined as differences between the observations and the projection of the forecasts onto the observation space. These methods are based on two main statistical criteria: 1) the method of moments, in which the theoretical and empirical moments of the innovations are assumed to be equal, and 2) methods that use the likelihood of the observations, themselves contained in the innovations. The reviewed methods assume that innovations are Gaussian random variables, although extension to other distributions is possible for likelihood-based methods. The methods also show some differences in terms of levels of complexity and applicability to high-dimensional systems. The conclusion of the review discusses the key challenges to further develop estimation methods for and . These challenges include taking into account time-varying error covariances, using limited observational coverage, estimating additional deterministic error terms, or accounting for correlated noise.

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Tài liệu tham khảo

Anderson, J. L., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210–224, https://doi.org/10.1111/j.1600-0870.2006.00216.x.

Anderson, J. L., 2009: Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus, 61A, 72–83, https://doi.org/10.1111/j.1600-0870.2008.00361.x.

Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741–2758, https://doi.org/10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

Bélanger, P. R., 1974: Estimation of noise covariance matrices for a linear time-varying stochastic process. Automatica, 10, 267–275, https://doi.org/10.1016/0005-1098(74)90037-5.

Berry, T., and T. Sauer, 2013: Adaptive ensemble Kalman filtering of non-linear systems. Tellus, 65A, 20331, https://doi.org/10.3402/tellusa.v65i0.20331.

Berry, T., and J. Harlim, 2017: Correcting biased observation model error in data assimilation. Mon. Wea. Rev., 145, 2833–2853, https://doi.org/10.1175/MWR-D-16-0428.1.

Berry, T., and T. Sauer, 2018: Correlation between system and observation errors in data assimilation. Mon. Wea. Rev., 146, 2913–2931, https://doi.org/10.1175/MWR-D-17-0331.1.

Bishop, C. H., and E. A. Satterfield, 2013: Hidden error variance theory. Part I: Exposition and analytic model. Mon. Wea. Rev., 141, 1454–1468, https://doi.org/10.1175/MWR-D-12-00118.1.

Bishop, C. H., E. A. Satterfield, and K. T. Shanley, 2013: Hidden error variance theory. Part II: An instrument that reveals hidden error variance distributions from ensemble forecasts and observations. Mon. Wea. Rev., 141, 1469–1483, https://doi.org/10.1175/MWR-D-12-00119.1.

Blanchet, I., C. Frankignoul, and M. A. Cane, 1997: A comparison of adaptive Kalman filters for a tropical Pacific Ocean model. Mon. Wea. Rev., 125, 40–58, https://doi.org/10.1175/1520-0493(1997)125<0040:ACOAKF>2.0.CO;2.

Bocquet, M., 2011: Ensemble Kalman filtering without the intrinsic need for inflation. Nonlinear Processes Geophys., 18, 735–750, https://doi.org/10.5194/npg-18-735-2011.

Bocquet, M., and P. Sakov, 2012: Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems. Nonlinear Processes Geophys., 19, 383–399, https://doi.org/10.5194/npg-19-383-2012.

Bocquet, M., P. N. Raanes, and A. Hannart, 2015: Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation. Nonlinear Processes Geophys., 22, 645–662, https://doi.org/10.5194/npg-22-645-2015.

Bormann, N., A. Collard, and P. Bauer, 2010: Estimates of spatial and interchannel observation-error characteristics for current sounder radiances for numerical weather prediction. II: Application to AIRS and IASI data. Quart. J. Roy. Meteor. Soc., 136, 1051–1063, https://doi.org/10.1002/qj.615.

Bowler, N. E., 2017: On the diagnosis of model error statistics using weak-constraint data assimilation. Quart. J. Roy. Meteor. Soc., 143, 1916–1928, https://doi.org/10.1002/qj.3051.

Brankart, J.-M., E. Cosme, C.-E. Testut, P. Brasseur, and J. Verron, 2010: Efficient adaptive error parameterizations for square root or ensemble Kalman filters: Application to the control of ocean mesoscale signals. Mon. Wea. Rev., 138, 932–950, https://doi.org/10.1175/2009MWR3085.1.

Buehner, M., 2010: Error statistics in data assimilation: Estimation and modelling. Data Assimilation: Making Sense of Observations, W. Lahoz, B. Khattatov, and R. Menard, Eds., Springer, 93–112.

Campbell, W. F., E. A. Satterfield, B. Ruston, and N. L. Baker, 2017: Accounting for correlated observation error in a dual-formulation 4D variational data assimilation system. Mon. Wea. Rev., 145, 1019–1032, https://doi.org/10.1175/MWR-D-16-0240.1.

Cappé, O., 2011: Online expectation-maximisation. Mixtures: Estimation and Applications, K. L. Mengersen, C. Robert, and M. Titterington, Eds.,Wiley Series in Probability and Statistics, John Wiley & Sons, 1–53.

Carrassi, A., M. Bocquet, L. Bertino, and G. Evensen, 2018: Data assimilation in the geosciences: An overview on methods, issues and perspectives. Wiley Interdiscip. Rev.: Climate Change, 9, e535, https://doi.org/10.1002/wcc.535.

Chapnik, B., G. Desroziers, F. Rabier, and O. Talagrand, 2004: Properties and first application of an error-statistics tuning method in variational assimilation. Quart. J. Roy. Meteor. Soc., 130, 2253–2275, https://doi.org/10.1256/qj.03.26.

Cocucci, T. J., M. Pulido, M. Lucini, and P. Tandeo, 2020: Model error covariance estimation in particle and ensemble Kalman filters using an online expectation-maximization algorithm. arXiv:2003.02109, https://arxiv.org/pdf/2003.02109.pdf.

Corazza, M., E. Kalnay, D. J. Patil, R. Morss, M. Cai, I. Szunyogh, B. R. Hunt, and J. A. Yorke, 2003: Use of the breeding technique to estimate the structure of the analysis “errors of the day.” Nonlinear Processes Geophys., 10, 233–243, https://doi.org/10.5194/npg-10-233-2003.

Daescu, D. N., and R. Todling, 2010: Adjoint sensitivity of the model forecast to data assimilation system error covariance parameters. Quart. J. Roy. Meteor. Soc., 136, 2000–2012, https://doi.org/10.1002/qj.693.

Daescu, D. N., and R. H. Langland, 2013: Error covariance sensitivity and impact estimation with adjoint 4D-Var: Theoretical aspects and first applications to NAVDAS-AR. Quart. J. Roy. Meteor. Soc., 139, 226–241, https://doi.org/10.1002/qj.1943.

Daley, R., 1991: Atmospheric Data Analysis. Cambridge University Press, 457 pp.

Daley, R., 1992: Estimating model-error covariances for application to atmospheric data assimilation. Mon. Wea. Rev., 120, 1735–1746, https://doi.org/10.1175/1520-0493(1992)120<1735:EMECFA>2.0.CO;2.

Dee, D. P., 1995: On-line estimation of error covariance parameters for atmospheric data assimilation. Mon. Wea. Rev., 123, 1128–1145, https://doi.org/10.1175/1520-0493(1995)123<1128:OLEOEC>2.0.CO;2.

Dee, D. P., 2005: Bias and data assimilation. Quart. J. Roy. Meteor. Soc., 131, 3323–3343, https://doi.org/10.1256/qj.05.137.

Dee, D. P., and A. M. da Silva, 1999: Maximum-likelihood estimation of forecast and observation error covariance parameters. Part I: Methodology. Mon. Wea. Rev., 127, 1822–1834, https://doi.org/10.1175/1520-0493(1999)127<1822:MLEOFA>2.0.CO;2.

Dee, D. P., S. E. Cohn, A. Dalcher, and M. Ghil, 1985: An efficient algorithm for estimating noise covariances in distributed systems. IEEE Trans. Autom. Control, 30, 1057–1065, https://doi.org/10.1109/TAC.1985.1103837.

Dee, D. P., G. Gaspari, C. Redder, L. Rukhovets, and A. M. da Silva, 1999: Maximum-likelihood estimation of forecast and observation error covariance parameters. Part II: Applications. Mon. Wea. Rev., 127, 1835–1849, https://doi.org/10.1175/1520-0493(1999)127<1835:MLEOFA>2.0.CO;2.

DelSole, T., and X. Yang, 2010: State and parameter estimation in stochastic dynamical models. Physica D, 239, 1781–1788, https://doi.org/10.1016/j.physd.2010.06.001.

Dempster, A. P., N. M. Laird, and D. B. Rubin, 1977: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc., 39B, 1–22, https://doi.org/10.1111/j.2517-6161.1977.tb01600.x.

Desroziers, G., and S. Ivanov, 2001: Diagnosis and adaptive tuning of observation-error parameters in a variational assimilation. Quart. J. Roy. Meteor. Soc., 127, 1433–1452, https://doi.org/10.1002/qj.49712757417.

Desroziers, G., L. Berre, B. Chapnik, and P. Poli, 2005: Diagnosis of observation, background and analysis-error statistics in observation space. Quart. J. Roy. Meteor. Soc., 131, 3385–3396, https://doi.org/10.1256/qj.05.108.

Dreano, D., P. Tandeo, M. Pulido, T. Chonavel, B. A. It-El-Fquih, and I. Hoteit, 2017: Estimating model error covariances in nonlinear state-space models using Kalman smoothing and the expectation-maximisation algorithm. Quart. J. Roy. Meteor. Soc., 143, 1877–1885, https://doi.org/10.1002/qj.3048.

Duník, J., O. Straka, O. Kost, and J. Havlík, 2017: Noise covariance matrices in state-space models: A survey and comparison of estimation methods-Part I. Int. J. Adapt. Control Signal Process., 31, 1505–1543, https://doi.org/10.1002/acs.2783.

El Gharamti, M., 2018: Enhanced adaptive inflation algorithm for ensemble filters. Mon. Wea. Rev., 146, 623–640, https://doi.org/10.1175/MWR-D-17-0187.1.

Evensen, G., 2009: Data Assimilation: The Ensemble Kalman Filter. Springer Science and Business Media, 332 pp.

Frei, M., and H. R. Künsch, 2012: Sequential state and observation noise covariance estimation using combined ensemble Kalman and particle filters. Mon. Wea. Rev., 140, 1476–1495, https://doi.org/10.1175/MWR-D-10-05088.1.

Fu, L.-L., I. Fukumori, and R. N. Miller, 1993: Fitting dynamic models to the Geosat sea level observations in the tropical Pacific Ocean. Part II: A linear, wind-driven model. J. Phys. Oceanogr., 23, 2162–2181, https://doi.org/10.1175/1520-0485(1993)023<2162:FDMTTG>2.0.CO;2.

Ghil, M., and P. Malanotte-Rizzoli, 1991: Data assimilation in meteorology and oceanography. Advances in Geophysics, Vol. 33, Academic Press, 141–266, https://doi.org/10.1016/S0065-2687(08)60442-2.

Guillet, O., A. T. Weaver, X. Vasseur, Y. Michel, S. Gratton, and S. Gürol, 2019: Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh. Quart. J. Roy. Meteor. Soc., 145, 1947–1967, https://doi.org/10.1002/qj.3537.

Harlim, J., 2018: Ensemble Kalman filters. Data-Driven Computational Methods: Parameter and Operator Estimations, J. Harlim, Ed., Cambridge University Press, 31–59.

Harlim, J., A. Mahdi, and A. J. Majda, 2014: An ensemble Kalman filter for statistical estimation of physics constrained nonlinear regression models. J. Comput. Phys., 257, 782–812, https://doi.org/10.1016/j.jcp.2013.10.025.

Hollingsworth, A., and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A, 111–136, https://doi.org/10.3402/tellusa.v38i2.11707.

Hotta, D., E. Kalnay, Y. Ota, and T. Miyoshi, 2017: EFSR: Ensemble forecast sensitivity to observation error covariance. Mon. Wea. Rev., 145, 5015–5031, https://doi.org/10.1175/MWR-D-17-0122.1.

Houtekamer, P. L., and F. Zhang, 2016: Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 144, 4489–4532, https://doi.org/10.1175/MWR-D-15-0440.1.

Houtekamer, P. L., H. L. Mitchell, and X. Deng, 2009: Model error representation in an operational ensemble Kalman filter. Mon. Wea. Rev., 137, 2126–2143, https://doi.org/10.1175/2008MWR2737.1.

Ide, K., P. Courtier, M. Ghil, and A. C. Lorenc, 1997: Unified notation for data assimilation: Operational, sequential and variational. J. Meteor. Soc. Japan, 75, 181–189, https://doi.org/10.2151/jmsj1965.75.1B_181.

Janjić, T., and Coauthors, 2018: On the representation error in data assimilation. Quart. J. Roy. Meteor. Soc., 144, 1257–1278, https://doi.org/10.1002/qj.3130.

Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

Kantas, N., A. Doucet, S. S. Singh, J. Maciejowski, and N. Chopin, 2015: On particle methods for parameter estimation in state-space models. Stat. Sci., 30, 328–351, https://doi.org/10.1214/14-STS511.

Katzfuss, M., J. R. Stroud, and C. K. Wikle, 2020: Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models. J. Amer. Stat. Assoc., 115, 866–885, https://doi.org/10.1080/01621459.2019.1592753.

Kotsuki, S., T. Miyoshi, K. Terasaki, G.-Y. Lien, and E. Kalnay, 2017a: Assimilating the global satellite mapping of precipitation data with the Nonhydrostatic Icosahedral Atmospheric Model (NICAM). J. Geophys. Res. Atmos., 122, 631–650, https://doi.org/10.1002/2016jd025355.

Kotsuki, S., Y. Ota, and T. Miyoshi, 2017b: Adaptive covariance relaxation methods for ensemble data assimilation: Experiments in the real atmosphere. Quart. J. Roy. Meteor. Soc., 143, 2001–2015, https://doi.org/10.1002/qj.3060.

Leith, C. E., 1978: Objective methods for weather prediction. Annu. Rev. Fluid Mech., 10, 107–128, https://doi.org/10.1146/annurev.fl.10.010178.000543.

Li, H., E. Kalnay, and T. Miyoshi, 2009: Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 135, 523–533, https://doi.org/10.1002/qj.371.

Liang, X., X. Zheng, S. Zhang, G. Wu, Y. Dai, and Y. Li, 2012: Maximum likelihood estimation of inflation factors on error covariance matrices for ensemble Kalman filter assimilation. Quart. J. Roy. Meteor. Soc., 138, 263–273, https://doi.org/10.1002/qj.912.

Liu, J., and M. West, 2001: Combined parameter and state estimation in simulation-based filtering. Sequential Monte Carlo Methods in Practice, A. Doucet, N. Freitas, and N. Gordon, Eds., Springer, 197–223.

Liu, Y., J.-M. Haussaire, M. Bocquet, Y. Roustan, O. Saunier, and A. Mathieu, 2017: Uncertainty quantification of pollutant source retrieval: Comparison of Bayesian methods with application to the Chernobyl and Fukushima Daiichi accidental releases of radionuclides. Quart. J. Roy. Meteor. Soc., 143, 2886–2901, https://doi.org/10.1002/qj.3138.

Mehra, R. K., 1970: On the identification of variances and adaptive Kalman filtering. IEEE Trans. Autom. Control, 15, 175–184, https://doi.org/10.1109/TAC.1970.1099422.

Mehra, R. K., 1972: Approaches to adaptive filtering. IEEE Trans. Autom. Control, 17, 693–698, https://doi.org/10.1109/TAC.1972.1100100.

Ménard, R., 2016: Error covariance estimation methods based on analysis residuals: Theoretical foundation and convergence properties derived from simplified observation networks. Quart. J. Roy. Meteor. Soc., 142, 257–273, https://doi.org/10.1002/qj.2650.

Menemenlis, D., and M. Chechelnitsky, 2000: Error estimates for an ocean general circulation model from altimeter and acoustic tomography data. Mon. Wea. Rev., 128, 763–778, https://doi.org/10.1175/1520-0493(2000)128<0763:EEFAOG>2.0.CO;2.

Ménétrier, B., and T. Auligné, 2015: Optimized localization and hybridization to filter ensemble-based covariances. Mon. Wea. Rev., 143, 3931–3947, https://doi.org/10.1175/MWR-D-15-0057.1.

Merchant, C. J., S. Saux-Picart, and J. Waller, 2020: Bias correction and covariance parameters for optimal estimation by exploiting matched in-situ references. Remote Sens. Environ., 237, 111590, https://doi.org/10.1016/j.rse.2019.111590.

Mitchell, H. L., and P. L. Houtekamer, 2000: An adaptive ensemble Kalman filter. Mon. Wea. Rev., 128, 416–433, https://doi.org/10.1175/1520-0493(2000)128<0416:AAEKF>2.0.CO;2.

Mitchell, L., and A. Carrassi, 2015: Accounting for model error due to unresolved scales within ensemble Kalman filtering. Quart. J. Roy. Meteor. Soc., 141, 1417–1428, https://doi.org/10.1002/qj.2451.

Miyoshi, T., 2011: The Gaussian approach to adaptive covariance inflation and its implementation with the local ensemble transform Kalman filter. Mon. Wea. Rev., 139, 1519–1535, https://doi.org/10.1175/2010MWR3570.1.

Miyoshi, T., Y. Sato, and T. Kadowaki, 2010: Ensemble Kalman filter and 4D-Var intercomparison with the Japanese operational global analysis and prediction system. Mon. Wea. Rev., 138, 2846–2866, https://doi.org/10.1175/2010MWR3209.1.

Miyoshi, T., E. Kalnay, and H. Li, 2013: Estimating and including observation-error correlations in data assimilation. Inverse Probl. Sci. Eng., 21, 387–398, https://doi.org/10.1080/17415977.2012.712527.

Pham, D. T., J. Verron, and M. C. Roubaud, 1998: A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst., 16, 323–340, https://doi.org/10.1016/S0924-7963(97)00109-7.

Pulido, M., P. Tandeo, M. Bocquet, A. Carrassi, and M. Lucini, 2018: Stochastic parameterization identification using ensemble Kalman filtering combined with maximum likelihood methods. Tellus, 70A, 1–17, https://doi.org/10.1080/16000870.2018.1442099.

Purser, R. J., and D. F. Parrish, 2003: A Bayesian technique for estimating continuously varying statistical parameters of a variational assimilation. Meteor. Atmos. Phys., 82, 209–226, https://doi.org/10.1007/s00703-001-0583-x.

Raanes, P. N., M. Bocquet, and A. Carrassi, 2019: Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures. Quart. J. Roy. Meteor. Soc., 145, 53–75, https://doi.org/10.1002/qj.3386.

Rodwell, M. J., and T. N. Palmer, 2007: Using numerical weather prediction to assess climate models. Quart. J. Roy. Meteor. Soc., 133, 129–146, https://doi.org/10.1002/qj.23.

Ruiz, J. J., M. Pulido, and T. Miyoshi, 2013a: Estimating model parameters with ensemble-based data assimilation: A review. J. Meteor. Soc. Japan, 91, 79–99, https://doi.org/10.2151/jmsj.2013-201.

Ruiz, J. J., M. Pulido, and T. Miyoshi, 2013b: Estimating model parameters with ensemble-based data assimilation: Parameter covariance treatment. J. Meteor. Soc. Japan, 91, 453–469, https://doi.org/10.2151/jmsj.2013-403.

Rutherford, I. D., 1972: Data assimilation by statistical interpolation of forecast error fields. J. Atmos. Sci., 29, 809–815, https://doi.org/10.1175/1520-0469(1972)029<0809:DABSIO>2.0.CO;2.

Satterfield, E. A., D. Hodyss, D. D. Kuhl, and C. H. Bishop, 2018: Observation-informed generalized hybrid error covariance models. Mon. Wea. Rev., 146, 3605–3622, https://doi.org/10.1175/MWR-D-18-0016.1.

Scheffler, G., J. Ruiz, and M. Pulido, 2019: Inference of stochastic parametrizations for model error treatment using nested ensemble Kalman filters. Quart. J. Roy. Meteor. Soc., 145, 2028–2045, https://doi.org/10.1002/qj.3542.

Schmidt, S. F., 1966: Applications of state space methods to navigation problems. Adv. Control Syst., 3, 293–340, https://doi.org/10.1016/B978-1-4831-6716-9.50011-4.

Shumway, R. H., and D. S. Stoffer, 1982: An approach to time series smoothing and forecasting using the EM algorithm. J. Time Ser. Anal., 3, 253–264, https://doi.org/10.1111/j.1467-9892.1982.tb00349.x.

Solonen, A., J. Hakkarainen, A. Ilin, M. Abbas, and A. Bibov, 2014: Estimating model error covariance matrix parameters in extended Kalman filtering. Nonlinear Processes Geophys., 21, 919–927, https://doi.org/10.5194/npg-21-919-2014.

Stroud, J. R., and T. Bengtsson, 2007: Sequential state and variance estimation within the ensemble Kalman filter. Mon. Wea. Rev., 135, 3194–3208, https://doi.org/10.1175/MWR3460.1.

Stroud, J. R., M. Katzfuss, and C. K. Wikle, 2018: A Bayesian adaptive ensemble Kalman filter for sequential state and parameter estimation. Mon. Wea. Rev., 146, 373–386, https://doi.org/10.1175/MWR-D-16-0427.1.

Tandeo, P., M. Pulido, and F. Lott, 2015: Offline parameter estimation using EnKF and maximum likelihood error covariance estimates: Application to a subgrid-scale orography parametrization. Quart. J. Roy. Meteor. Soc., 141, 383–395, https://doi.org/10.1002/qj.2357.

Todling, R., 2015: A lag-1 smoother approach to system-error estimation: Sequential method. Quart. J. Roy. Meteor. Soc., 141, 1502–1513, https://doi.org/10.1002/qj.2460.

Ueno, G., and N. Nakamura, 2014: Iterative algorithm for maximum-likelihood estimation of the observation-error covariance matrix for ensemble-based filters. Quart. J. Roy. Meteor. Soc., 140, 295–315, https://doi.org/10.1002/qj.2134.

Ueno, G., and N. Nakamura, 2016: Bayesian estimation of the observation-error covariance matrix in ensemble-based filters. Quart. J. Roy. Meteor. Soc., 142, 2055–2080, https://doi.org/10.1002/qj.2803.

Ueno, G., T. Higuchi, T. Kagimoto, and N. Hirose, 2010: Maximum likelihood estimation of error covariances in ensemble-based filters and its application to a coupled atmosphere-ocean model. Quart. J. Roy. Meteor. Soc., 136, 1316–1343, https://doi.org/10.1002/qj.654.

Wahba, G., and J. Wendelberger, 1980: Some new mathematical methods for variational objective analysis using splines and cross validation. Mon. Wea. Rev., 108, 1122–1143, https://doi.org/10.1175/1520-0493(1980)108<1122:SNMMFV>2.0.CO;2.

Waller, J. A., S. L. Dance, and N. K. Nichols, 2016: Theoretical insight into diagnosing observation error correlations using observation-minus-background and observation-minus-analysis statistics. Quart. J. Roy. Meteor. Soc., 142, 418–431, https://doi.org/10.1002/qj.2661.

Waller, J. A., S. L. Dance, and N. K. Nichols, 2017: On diagnosing observation-error statistics with local ensemble data assimilation. Quart. J. Roy. Meteor. Soc., 143, 2677–2686, https://doi.org/10.1002/qj.3117.

Wang, X., and C. H. Bishop, 2003: A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes. J. Atmos. Sci., 60, 1140–1158, https://doi.org/10.1175/1520-0469(2003)060<1140:ACOBAE>2.0.CO;2.

Weston, P. P., W. Bell, and J. R. Eyre, 2014: Accounting for correlated error in the assimilation of high-resolution sounder data. Quart. J. Roy. Meteor. Soc., 140, 2420–2429, https://doi.org/10.1002/qj.2306.

Whitaker, J. S., and T. M. Hamill, 2012: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 3078–3089, https://doi.org/10.1175/MWR-D-11-00276.1.

Whitaker, J. S., T. M. Hamill, X. Wei, Y. Song, and Z. Toth, 2008: Ensemble data assimilation with the NCEP global forecast system. Mon. Wea. Rev., 136, 463–482, https://doi.org/10.1175/2007MWR2018.1.

Winiarek, V., M. Bocquet, O. Saunier, and A. Mathieu, 2012: Estimation of errors in the inverse modeling of accidental release of atmospheric pollutant: Application to the reconstruction of the cesium-137 and iodine-131 source terms from the Fukushima Daiichi power plant. J. Geophys. Res., 117, D05122, https://doi.org/10.1029/2011JD016932.

Winiarek, V., M. Bocquet, N. Duhanyan, Y. Roustan, O. Saunier, and A. Mathieu, 2014: Estimation of the caesium-137 source term from the Fukushima Daiichi nuclear power plant using a consistent joint assimilation of air concentration and deposition observations. Atmos. Environ., 82, 268–279, https://doi.org/10.1016/j.atmosenv.2013.10.017.

Wu, C. F. J., 1983: On the convergence properties of the EM algorithm. Ann. Stat., 11, 95–103, https://doi.org/10.1214/aos/1176346060.

Yang, Y., and E. Mémin, 2019: Estimation of physical parameters under location uncertainty using an ensemble-expectation-maximization algorithm. Quart. J. Roy. Meteor. Soc., 145, 418–433, https://doi.org/10.1002/qj.3438.

Ying, Y., and F. Zhang, 2015: An adaptive covariance relaxation method for ensemble data assimilation. Quart. J. Roy. Meteor. Soc., 141, 2898–2906, https://doi.org/10.1002/qj.2576.

Zebiak, S. E., and M. A. Cane, 1987: A model El Niño–Southern Oscillation. Mon. Wea. Rev., 115, 2262–2278, https://doi.org/10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2.

Zhang, F., C. Snyder, and J. Sun, 2004: Impacts of initial estimate and observation availability on convective-scale data assimilation with an ensemble Kalman filter. Mon. Wea. Rev., 132, 1238–1253, https://doi.org/10.1175/1520-0493(2004)132<1238:IOIEAO>2.0.CO;2.

Zhen, Y., and J. Harlim, 2015: Adaptive error covariances estimation methods for ensemble Kalman filters. J. Comput. Phys., 294, 619–638, https://doi.org/10.1016/j.jcp.2015.03.061.

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Monthly Weather Review

Tập 148 Số 10

3973-3994

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